Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [85,2,Mod(21,85)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("85.21"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 85.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.678728417181\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{10} + 83x^{8} + 152x^{6} + 111x^{4} + 22x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 21.6
Root \(0.254679i\) of defining polynomial
Character \(\chi\) \(=\) 85.21
Dual form 85.2.e.a.81.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.51230i q^{2} +(-0.887192 + 0.887192i) q^{3} -4.31167 q^{4} +(0.707107 - 0.707107i) q^{5} +(-2.22889 - 2.22889i) q^{6} +(1.14187 + 1.14187i) q^{7} -5.80761i q^{8} +1.42578i q^{9} +(1.77647 + 1.77647i) q^{10} +(-2.32404 - 2.32404i) q^{11} +(3.82528 - 3.82528i) q^{12} +6.35524 q^{13} +(-2.86872 + 2.86872i) q^{14} +1.25468i q^{15} +5.96715 q^{16} +(-0.768287 + 4.05089i) q^{17} -3.58200 q^{18} -0.747167i q^{19} +(-3.04881 + 3.04881i) q^{20} -2.02612 q^{21} +(5.83869 - 5.83869i) q^{22} +(0.101240 + 0.101240i) q^{23} +(5.15247 + 5.15247i) q^{24} -1.00000i q^{25} +15.9663i q^{26} +(-3.92652 - 3.92652i) q^{27} +(-4.92337 - 4.92337i) q^{28} +(6.22477 - 6.22477i) q^{29} -3.15213 q^{30} +(5.10259 - 5.10259i) q^{31} +3.37606i q^{32} +4.12374 q^{33} +(-10.1771 - 1.93017i) q^{34} +1.61485 q^{35} -6.14750i q^{36} +(-0.439195 + 0.439195i) q^{37} +1.87711 q^{38} +(-5.63832 + 5.63832i) q^{39} +(-4.10660 - 4.10660i) q^{40} +(-4.49889 - 4.49889i) q^{41} -5.09022i q^{42} +2.74801i q^{43} +(10.0205 + 10.0205i) q^{44} +(1.00818 + 1.00818i) q^{45} +(-0.254346 + 0.254346i) q^{46} -11.9308 q^{47} +(-5.29400 + 5.29400i) q^{48} -4.39226i q^{49} +2.51230 q^{50} +(-2.91230 - 4.27554i) q^{51} -27.4017 q^{52} +2.71404i q^{53} +(9.86460 - 9.86460i) q^{54} -3.28669 q^{55} +(6.63154 - 6.63154i) q^{56} +(0.662880 + 0.662880i) q^{57} +(15.6385 + 15.6385i) q^{58} +12.6815i q^{59} -5.40976i q^{60} +(0.328162 + 0.328162i) q^{61} +(12.8193 + 12.8193i) q^{62} +(-1.62806 + 1.62806i) q^{63} +3.45261 q^{64} +(4.49384 - 4.49384i) q^{65} +10.3601i q^{66} +1.81686 q^{67} +(3.31260 - 17.4661i) q^{68} -0.179639 q^{69} +4.05699i q^{70} +(-3.01043 + 3.01043i) q^{71} +8.28039 q^{72} +(-0.856488 + 0.856488i) q^{73} +(-1.10339 - 1.10339i) q^{74} +(0.887192 + 0.887192i) q^{75} +3.22154i q^{76} -5.30750i q^{77} +(-14.1652 - 14.1652i) q^{78} +(-3.57000 - 3.57000i) q^{79} +(4.21941 - 4.21941i) q^{80} +2.68980 q^{81} +(11.3026 - 11.3026i) q^{82} -3.58494i q^{83} +8.73594 q^{84} +(2.32115 + 3.40767i) q^{85} -6.90384 q^{86} +11.0451i q^{87} +(-13.4971 + 13.4971i) q^{88} -10.5906 q^{89} +(-2.53285 + 2.53285i) q^{90} +(7.25687 + 7.25687i) q^{91} +(-0.436515 - 0.436515i) q^{92} +9.05395i q^{93} -29.9739i q^{94} +(-0.528327 - 0.528327i) q^{95} +(-2.99521 - 2.99521i) q^{96} +(-4.92762 + 4.92762i) q^{97} +11.0347 q^{98} +(3.31357 - 3.31357i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} - 12 q^{4} - 4 q^{10} - 4 q^{11} - 8 q^{12} - 4 q^{14} + 4 q^{16} + 12 q^{17} + 28 q^{18} - 8 q^{20} - 16 q^{21} + 20 q^{22} + 12 q^{23} + 4 q^{24} - 4 q^{27} + 4 q^{28} - 12 q^{29} - 8 q^{30}+ \cdots + 44 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(71\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.51230i 1.77647i 0.459392 + 0.888233i \(0.348067\pi\)
−0.459392 + 0.888233i \(0.651933\pi\)
\(3\) −0.887192 + 0.887192i −0.512220 + 0.512220i −0.915206 − 0.402986i \(-0.867972\pi\)
0.402986 + 0.915206i \(0.367972\pi\)
\(4\) −4.31167 −2.15583
\(5\) 0.707107 − 0.707107i 0.316228 − 0.316228i
\(6\) −2.22889 − 2.22889i −0.909943 − 0.909943i
\(7\) 1.14187 + 1.14187i 0.431586 + 0.431586i 0.889168 − 0.457581i \(-0.151284\pi\)
−0.457581 + 0.889168i \(0.651284\pi\)
\(8\) − 5.80761i − 2.05330i
\(9\) 1.42578i 0.475261i
\(10\) 1.77647 + 1.77647i 0.561768 + 0.561768i
\(11\) −2.32404 − 2.32404i −0.700724 − 0.700724i 0.263842 − 0.964566i \(-0.415010\pi\)
−0.964566 + 0.263842i \(0.915010\pi\)
\(12\) 3.82528 − 3.82528i 1.10426 − 1.10426i
\(13\) 6.35524 1.76263 0.881314 − 0.472531i \(-0.156660\pi\)
0.881314 + 0.472531i \(0.156660\pi\)
\(14\) −2.86872 + 2.86872i −0.766699 + 0.766699i
\(15\) 1.25468i 0.323957i
\(16\) 5.96715 1.49179
\(17\) −0.768287 + 4.05089i −0.186337 + 0.982486i
\(18\) −3.58200 −0.844284
\(19\) − 0.747167i − 0.171412i −0.996320 − 0.0857059i \(-0.972685\pi\)
0.996320 − 0.0857059i \(-0.0273145\pi\)
\(20\) −3.04881 + 3.04881i −0.681735 + 0.681735i
\(21\) −2.02612 −0.442135
\(22\) 5.83869 − 5.83869i 1.24481 − 1.24481i
\(23\) 0.101240 + 0.101240i 0.0211101 + 0.0211101i 0.717583 − 0.696473i \(-0.245248\pi\)
−0.696473 + 0.717583i \(0.745248\pi\)
\(24\) 5.15247 + 5.15247i 1.05174 + 1.05174i
\(25\) − 1.00000i − 0.200000i
\(26\) 15.9663i 3.13125i
\(27\) −3.92652 − 3.92652i −0.755659 − 0.755659i
\(28\) −4.92337 − 4.92337i −0.930429 − 0.930429i
\(29\) 6.22477 − 6.22477i 1.15591 − 1.15591i 0.170565 − 0.985346i \(-0.445441\pi\)
0.985346 − 0.170565i \(-0.0545592\pi\)
\(30\) −3.15213 −0.575498
\(31\) 5.10259 − 5.10259i 0.916452 − 0.916452i −0.0803174 − 0.996769i \(-0.525593\pi\)
0.996769 + 0.0803174i \(0.0255934\pi\)
\(32\) 3.37606i 0.596809i
\(33\) 4.12374 0.717850
\(34\) −10.1771 − 1.93017i −1.74535 − 0.331021i
\(35\) 1.61485 0.272959
\(36\) − 6.14750i − 1.02458i
\(37\) −0.439195 + 0.439195i −0.0722032 + 0.0722032i −0.742286 − 0.670083i \(-0.766258\pi\)
0.670083 + 0.742286i \(0.266258\pi\)
\(38\) 1.87711 0.304507
\(39\) −5.63832 + 5.63832i −0.902854 + 0.902854i
\(40\) −4.10660 − 4.10660i −0.649311 − 0.649311i
\(41\) −4.49889 − 4.49889i −0.702609 − 0.702609i 0.262361 − 0.964970i \(-0.415499\pi\)
−0.964970 + 0.262361i \(0.915499\pi\)
\(42\) − 5.09022i − 0.785438i
\(43\) 2.74801i 0.419068i 0.977801 + 0.209534i \(0.0671947\pi\)
−0.977801 + 0.209534i \(0.932805\pi\)
\(44\) 10.0205 + 10.0205i 1.51064 + 1.51064i
\(45\) 1.00818 + 1.00818i 0.150291 + 0.150291i
\(46\) −0.254346 + 0.254346i −0.0375013 + 0.0375013i
\(47\) −11.9308 −1.74029 −0.870146 − 0.492794i \(-0.835976\pi\)
−0.870146 + 0.492794i \(0.835976\pi\)
\(48\) −5.29400 + 5.29400i −0.764124 + 0.764124i
\(49\) − 4.39226i − 0.627466i
\(50\) 2.51230 0.355293
\(51\) −2.91230 − 4.27554i −0.407804 − 0.598695i
\(52\) −27.4017 −3.79993
\(53\) 2.71404i 0.372802i 0.982474 + 0.186401i \(0.0596823\pi\)
−0.982474 + 0.186401i \(0.940318\pi\)
\(54\) 9.86460 − 9.86460i 1.34240 − 1.34240i
\(55\) −3.28669 −0.443177
\(56\) 6.63154 − 6.63154i 0.886177 − 0.886177i
\(57\) 0.662880 + 0.662880i 0.0878006 + 0.0878006i
\(58\) 15.6385 + 15.6385i 2.05344 + 2.05344i
\(59\) 12.6815i 1.65099i 0.564413 + 0.825493i \(0.309103\pi\)
−0.564413 + 0.825493i \(0.690897\pi\)
\(60\) − 5.40976i − 0.698397i
\(61\) 0.328162 + 0.328162i 0.0420168 + 0.0420168i 0.727803 − 0.685786i \(-0.240542\pi\)
−0.685786 + 0.727803i \(0.740542\pi\)
\(62\) 12.8193 + 12.8193i 1.62805 + 1.62805i
\(63\) −1.62806 + 1.62806i −0.205116 + 0.205116i
\(64\) 3.45261 0.431576
\(65\) 4.49384 − 4.49384i 0.557392 − 0.557392i
\(66\) 10.3601i 1.27524i
\(67\) 1.81686 0.221965 0.110982 − 0.993822i \(-0.464600\pi\)
0.110982 + 0.993822i \(0.464600\pi\)
\(68\) 3.31260 − 17.4661i 0.401712 − 2.11808i
\(69\) −0.179639 −0.0216260
\(70\) 4.05699i 0.484903i
\(71\) −3.01043 + 3.01043i −0.357272 + 0.357272i −0.862806 − 0.505535i \(-0.831295\pi\)
0.505535 + 0.862806i \(0.331295\pi\)
\(72\) 8.28039 0.975853
\(73\) −0.856488 + 0.856488i −0.100244 + 0.100244i −0.755450 − 0.655206i \(-0.772582\pi\)
0.655206 + 0.755450i \(0.272582\pi\)
\(74\) −1.10339 − 1.10339i −0.128267 − 0.128267i
\(75\) 0.887192 + 0.887192i 0.102444 + 0.102444i
\(76\) 3.22154i 0.369535i
\(77\) − 5.30750i − 0.604846i
\(78\) −14.1652 − 14.1652i −1.60389 − 1.60389i
\(79\) −3.57000 − 3.57000i −0.401656 − 0.401656i 0.477160 − 0.878816i \(-0.341666\pi\)
−0.878816 + 0.477160i \(0.841666\pi\)
\(80\) 4.21941 − 4.21941i 0.471744 − 0.471744i
\(81\) 2.68980 0.298867
\(82\) 11.3026 − 11.3026i 1.24816 − 1.24816i
\(83\) − 3.58494i − 0.393498i −0.980454 − 0.196749i \(-0.936962\pi\)
0.980454 − 0.196749i \(-0.0630385\pi\)
\(84\) 8.73594 0.953169
\(85\) 2.32115 + 3.40767i 0.251764 + 0.369614i
\(86\) −6.90384 −0.744460
\(87\) 11.0451i 1.18416i
\(88\) −13.4971 + 13.4971i −1.43880 + 1.43880i
\(89\) −10.5906 −1.12260 −0.561300 − 0.827613i \(-0.689698\pi\)
−0.561300 + 0.827613i \(0.689698\pi\)
\(90\) −2.53285 + 2.53285i −0.266986 + 0.266986i
\(91\) 7.25687 + 7.25687i 0.760726 + 0.760726i
\(92\) −0.436515 − 0.436515i −0.0455098 − 0.0455098i
\(93\) 9.05395i 0.938851i
\(94\) − 29.9739i − 3.09157i
\(95\) −0.528327 − 0.528327i −0.0542052 − 0.0542052i
\(96\) −2.99521 − 2.99521i −0.305698 − 0.305698i
\(97\) −4.92762 + 4.92762i −0.500324 + 0.500324i −0.911539 − 0.411215i \(-0.865105\pi\)
0.411215 + 0.911539i \(0.365105\pi\)
\(98\) 11.0347 1.11467
\(99\) 3.31357 − 3.31357i 0.333026 − 0.333026i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 85.2.e.a.21.6 ✓ 12
3.2 odd 2 765.2.k.b.361.1 12
4.3 odd 2 1360.2.bt.d.1041.4 12
5.2 odd 4 425.2.j.b.174.1 12
5.3 odd 4 425.2.j.c.174.6 12
5.4 even 2 425.2.e.f.276.1 12
17.2 even 8 1445.2.d.g.866.2 12
17.8 even 8 1445.2.a.o.1.6 6
17.9 even 8 1445.2.a.n.1.6 6
17.13 even 4 inner 85.2.e.a.81.1 yes 12
17.15 even 8 1445.2.d.g.866.1 12
51.47 odd 4 765.2.k.b.676.6 12
68.47 odd 4 1360.2.bt.d.81.4 12
85.9 even 8 7225.2.a.bb.1.1 6
85.13 odd 4 425.2.j.b.149.1 12
85.47 odd 4 425.2.j.c.149.6 12
85.59 even 8 7225.2.a.z.1.1 6
85.64 even 4 425.2.e.f.251.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.e.a.21.6 ✓ 12 1.1 even 1 trivial
85.2.e.a.81.1 yes 12 17.13 even 4 inner
425.2.e.f.251.6 12 85.64 even 4
425.2.e.f.276.1 12 5.4 even 2
425.2.j.b.149.1 12 85.13 odd 4
425.2.j.b.174.1 12 5.2 odd 4
425.2.j.c.149.6 12 85.47 odd 4
425.2.j.c.174.6 12 5.3 odd 4
765.2.k.b.361.1 12 3.2 odd 2
765.2.k.b.676.6 12 51.47 odd 4
1360.2.bt.d.81.4 12 68.47 odd 4
1360.2.bt.d.1041.4 12 4.3 odd 2
1445.2.a.n.1.6 6 17.9 even 8
1445.2.a.o.1.6 6 17.8 even 8
1445.2.d.g.866.1 12 17.15 even 8
1445.2.d.g.866.2 12 17.2 even 8
7225.2.a.z.1.1 6 85.59 even 8
7225.2.a.bb.1.1 6 85.9 even 8